Laplace Transform of Real Power

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Theorem

Let $n$ be a constant real number such that $n > -1$

Let $f: \R \to \R$ be the real function defined as:

$\map f t = t^n$


Then $f$ has a Laplace transform given by:

$\laptrans {\map f t} = \dfrac {\map \Gamma {n + 1} } {s^{n + 1} }$

where $\Gamma$ denotes the gamma function.


Proof

\(\displaystyle \laptrans {t^n}\) \(=\) \(\displaystyle \int_0^\infty e^{-s t} t^n \rd t\) Definition of Laplace Transform
\(\displaystyle \) \(=\) \(\displaystyle \int_0^\infty e^{-u} \paren {\dfrac u s}^n \rd \paren {\dfrac u s}\) Integration by Substitution: $u := s t$ where $s > 0$ is assumed
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {s^{n + 1} } \int_0^\infty u^n e^{-u} \rd u\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\map \Gamma {n + 1} } {s^{n + 1} }\) Definition of Gamma Function

$\blacksquare$


Sources